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1、耦合非线性薛定谔方程多孤子解及其传输特性研究 目 录中 文 摘 要IABSTRACTII第一章 绪论11.1. 引言11.2. 非线性薛定谔方程的研究进展11.3. 孤子的研究进展21.4. 呼吸子的研究进展31.5. 本文的主要内容4第二章 GCNLSE的4-亮-亮孤子解及相互作用62.1. 引言62.2. 4-亮-亮孤子解72.3. 孤子间的相互作用122.3.1. ki (i = 1, 2, 3, 4)全为复数122.3.2. k1, k2实数和k3, k4复数142.4. 本章小结18第三章 GCNLSE 的4-暗-暗孤子解及相互作用193.1. 引言193.2. 4-暗-暗孤子解19
2、3.3. 4-暗孤子解的传输特性213.4. 本章小结24第四章 变系数CNLSE的二孤子解及相互作用254.1. 引言254.2. 变系数CNLSE254.3. 二孤子解264.4. 孤子相互作用274.4.1. 常规态二孤子间的相互作用284.4.2. 束缚态二孤子间的相互作用334.5. 本章小结36第五章 总结和展望385.1. 总结385.2. 展望39参 考 文 献40致谢49攻读学位期间取得的研究成果50个人简介及联系方式51承诺书52学位论文使用授权声明53ContentsChinese AbstractIABSTRACTIIChapter 1 Introduction11.1
3、. Introduction11.2. Research progress of nonlinear Schrdinger equation11.3. Research progress of solitons21.4. Research progress of breather31.5. Main research contents of the paper4Chapter 2 Four-bright-bright soliton solution and interactions of the GCNLSE62.1. Introduction62.2. Four-bright-bright
4、 soliton solution72.3. Soliton interactions122.3.1. ki (i = 1, 2, 3, 4) are all complex122.3.2. k1, k2 are real and k3, k4 are complex142.4. Summary of this chapter18Chapter 3 Four-dark-dark soliton solution and soliton interactions of the GCNLSE193.1. Introduction193.2. Four-dark-dark soliton solut
5、ion193.3. Interactions between solitons213.4. Summary of this chapter24Chapter 4 Two-soliton solution and interaction of variable coefficient CNLSE254.1. Introduction254.2. Variable coefficient CNLSE254.3. Two-soliton solution264.4. Interactions between solitons274.4.1. Interaction between two regul
6、ar state solitons284.4.2. Interaction between two bounded state solitons334.5. Summary of this chapter36Chapter 5 Summary and expectation385.1. Summary385.2. Expectation39References40Acknowledgment49Research achievement50Personal profiles51Letter of commitment52Authorization statement53中 文 摘 要由于群速度色
7、散和自相位调制之间的相互平衡，光孤子可以在光纤中长距离传输且形状不发生改变，因为这一特性，孤子可以在光纤通信系统中实现远距离和大容量传输，并可以应用在很多领域当中，成为了很多学者研究的内容。非线性薛定谔方程是描述光孤子传输的理想模型，是一类非常重要的非线性演化方程。随着研究的进行，非线性薛定谔方程被推广到了变系数、复系数、多维、高阶、非局域和分数阶等包含各类物理效应的方程，通过对各种方程解的研究可以更好地理解不同的非线性现象。因此，基于非线性薛定谔方程研究孤子的传输特性以及潜在的一些应用是至关重要的，对孤子在不同应用领域的发展具有一定的理论指导意义。本文主要介绍了非线性薛定谔方程的研究背景和进
8、展，孤子和呼吸子的由来和研究进展，在此基础上，采用Hirota双线性方法研究了多孤子解及其相互作用，具体的研究内容分为以下三个部分：（1） 基于自聚焦广义耦合非线性薛定谔方程，其中包含自相位调制、交叉相位调制和四波混频效应，采用Hirota双线性方法得到了该方程的4-亮-亮孤子解，并对孤子的碰撞动力学进行了详细地研究。研究结果表明：特征值的虚部影响孤子的速度和脉冲宽度，而孤子的振幅主要有特征值的实部决定。（2） 基于包含四波混频效应的自散焦广义耦合非线性薛定谔方程，采用Hirota双线性方法得到了该方程的4-暗-暗孤子解，分析不同参数的取值范围，数值研究了其传输特性。研究结果表明：通过调控参数
9、，可以获得三种不同的情况，即4-暗-暗孤子、3-暗-暗孤子及暗孤子-反暗孤子的组合，相互作用均表现为弹性碰撞。（3） 基于变系数耦合非线性薛定谔方程，利用Hirota双线性方法得到了该方程三亮一暗的四分量2-孤子解。孤子特征值取复数时，通过计算给出满足弹性碰撞的两种条件，一种为常规的弹性碰撞，另一种为一个亮孤子消失的准弹性碰撞。当孤子特征值取常数时，出现束缚态孤子。因此，通过合理选择参数，可以获得二孤子解的弹性碰撞、非弹性碰撞和束缚态传输等情况。关键词： 耦合非线性薛定谔方程； 孤子解； Hirota双线性方法；ABSTRACTDue to the balance between group
10、velocity dispersion and self-frequency shift modulation, optical solitons can travel long distances in the fiber without changing the shape. Because of this characteristic, solitons can be transmitted over large distances and large capacity in optical fiber communication systems. It can be applied i
11、n many fields and has become the content of many scholars research. The nonlinear Schrdinger equation is an ideal model for describing the soliton transmission. It is a very important nonlinear evolution equation. As the research progresses, the nonlinear Schrdinger equation is extended to variable
12、coefficients, complex coefficients, high-dimensional, high-order, non-local and fractional equations include various types of physical effects. By studying various equations, we can better understand many nonlinear phenomena. Therefore, it is very important to study the transmission characteristics
13、of soliton and some potential applications through the nonlinear Schrdinger equation. It has certain theoretical guiding significance for the development of soliton theory and the development of different application fields.This article mainly introduces the research background and progress of nonli
14、near Schrdinger equation, the origin and research progress of solitons and breathers. Based on these, the N-soltion solutions and their interactions are studied by using the Hirota bilinear method. The specific research contents are as follows:(1) Based on the self-focusing generalized coupled nonli
15、near Schrdinger equation, which includes self-phase modulation, cross-phase modulation, and four-wave mixing effects, the 4-bright-bright soliton solution of the equation is obtained by using the Hirota bilinear method. The soliton collision dynamics have been studied in detail. The results show that the imaginary part of the eigenvalue affects the speed and pulse width of the soliton, and the real part of the eigenvalue change the amplitude of the soliton.(2) Based on the self-de
